Kazuhiro Hikami
We consider an asymptotic expansion of Kashaev's invariant or the colored Jones function for the torus link T(2,2m). We shall give q-series identity related to these invariants, and show that the invariant is regarded as a limit of q being N-th root of unity of the Eichler integral of the modular form of weight 3/2.
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The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2) character manifold of the peripheral torus. We compute the asymptotics of the torus knot states in terms of the Alexander polynomial, the Reidemeister torsion and the Chern-Simons invariant. We also prove that the microsupport of the torus knot state is included in the character manifold of the knot exterior. As a corollary we deduce the Witten asymptotics conjecture for the Dehn filling of the torus knots and asymptotic expansions for the colored Jones polynomials…
We propose a method to compute complex volume of 2-bridge link complements. Our construction sheds light on a relationship between cluster variables with coefficients and canonical decompositions of link complements.
A relation between the two-variable series knot invariant and the Akutus-Deguchi-Ohtsuki(ADO)-invariant was conjectured recently. We reinforce the conjecture by presenting explicit formulas and/or an algorithm for certain ADO-invariants of torus knots obtained from the series invariant of complement of a knot. Furthermore, one parameter deformation of ADO_3-polynomial of torus knots is provided.
In this paper we prove a mirror symmetry conjecture based on the work of Brini-Eynard-Mariño \cite{BEM} and Diaconescu-Shende-Vafa \cite{DSV}. This conjecture relates open Gromov-Witten invariants of the conifold transition of a torus knot to the topological recursion on the B-model spectral curve.
Torus Knots in Adjoint Representation
本文推导了环形结的adjoint多项式闭式表达式，并探讨了其特殊性质，为Chern-Simons理论的普遍性提供了新的见解。
We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar-Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.
In 1993 Rosso and Jones computed for every simple, complex Lie algebra g_C and every colored torus knot in S^3 the value of the corresponding U_q(g_C)-quantum invariant by using the machinery of quantum groups. In the present paper we derive a S^2 x S^1-analogue of the Rosso-Jones formula (for colored torus ribbon knots) directly from a rigorous realization of the corresponding (gauge fixed) Chern-Simons path integral. In order to compare the explicit expressions obtained for torus knots in S^2 x S^1 with those for torus knots in S^3 one can perform a suitable surgery operation. By doing so we…
Gang-Kim-Yoon integrality conjectures on adjoint Reidemeister torsions for torus knots
本文研究了 torus 结绳的 adjoint Reidemeister 级数的整性猜想，通过引入 torus 结绳的 Verlinde 数并证明其递推公式和初始值，从而证明了该级数的整性。
Simple physics ideas are used to derive an exact expression for a flat connection on the complement of a torus knot. The result is of some mathematical importance in the context of constructing representations of the knot group -- a topological invariant of the knot. It is also a step forward in the direction of obtaining a generalisation of the Aharonov-Bohm effect, in which charged particles moving through force-free regions are scattered by impenetrable, knotted solenoids.
We briefly review the instanton-torus knot duality found in 5d SUSY gauge theories with one compact dimension. The fermion condensate turn out to be the generating function for the torus knot polynomials colored by the fundamental representation.
I consider the quantum interference of electrons moving along knotted trajectories under external magnetic field. The induced persistent current is formulated in terms of characteristic parameters that classify the torus knot geometry. The current is found to show a periodic oscillation whose period depends strongly both on the knot's class and the field direction. The shift in the oscillation period caused by geometric distortion is also discussed.
The explicit formula, which expresses the Alexander polynomials Δ_{n,3}(t) of torus knots T(n,3) as a sum of the Alexander polynomials Δ_{k,2}(t) of torus knots T(k,2), is found. Using this result and those from our previous papers, we express the Alexander polynomials Δ_{n,3}(t) through Chebyshev polynomials. The latter result is extended to general torus knots T(n,l) with n and l coprime.
Large-$N$ Torus Knots in Lens Spaces and Their Quiver Structure
本文研究了在Lens空间S³/Z_p中的torus结不变量，通过手术和模描述推导出一般表达式，并在大N极限下得到通用形式，通过重新定义结变量，生成函数表现出类似Quiver分区函数的结构，从而确定了torus结在Lens空间中的Quiver结构。
Augmentation curve for torus knots
该研究证明了环面纽结着色HOMFLY-PT多项式相关微分在增广曲线上的拓扑递归版本，构造并验证了谱曲线与增广曲线的一致性，证实了Aganagic-Vafa猜想的高亏格闭弦预测。
We construct Virasoro algebra of differential operators for the Jones-Rosso matrix model. These operators generate various relations between Wilson loops. Then we discuss the con- structed operators and corresponding relations in the stability limit. A series of examples is given.
We have studied the dynamics and symmetries of a particle constrained to move in a torus knot. The Hamiltonian system turns out to be Second Class in Dirac's formulation and the Dirac brackets yield novel noncommutative structures. The equations of motion are obtained for a path in general where the knot is present in the particle orbit but it is not restricted to a particular torus. We also study the motion when it is restricted to a specific torus. The rotational symmetries are studied as well. We have also considered the behavior of small fluctuations of the particle motion about a fixed to…
In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely connected with the Chebyshev polynomials, can also be related with the Alexander polynomials for the class T(s,2) of torus knots, s being an odd integer, and used for finding the corresponding skein relation. Then, we develop this procedure in order to obtain, with the help of q,p-numbers, the generalized two-variable Alexander polynomials, and prove their direct c…
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